Let $n$ be a positive integer (assume $n$ is prime for simplicity), and let $x_k = \pm1$, for $k = 0,1,2,..., n-1$. Let $\rho$ be an $n-$th primitive root of unity, I am interested in a lower bound for the absolute value of all possible sums of the form:

$$S_n = \sum_{k=0}^{n-1} x_k \rho^k$$ assuming that such sum is not equal to zero (when $n$ is prime this can only happen when all the $x_k$'s have the same sign).

One can obtain an easy lower bound of $\frac{1}{n^{n-1}}$ by multiplying the algebraic integer $S_n$ by all its Galois conjugates, but given that there are $2^n$ possible $S_n$ contained in a ball of radius $n$, I am expecting a better lower bound (hopefully $e^{-Cn}$ for some constant $C$).

I am also interested in the probability $$Pr(|S_n| < e^{-100n})$$, where $100$ is arbitrary. I am expecting this quantity to be exponentially small, I think from an argument in Tao-Vu's paper (https://arxiv.org/abs/1307.4357) related to Nguyen-Vu's optimal Offord-Littlewood inverse Theorem one might be able to show that such probability is smaller than $n^{-C}$ (for any fixed $C$ and $n \to \infty$), but I'm still far from understanding their method.

I would be grateful for any information related to sums of this form, similar sums or some understanding in how difficult this question can be.

Thanks!

1more comment